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The Boussinesq approximation is applied to problems where the fluid varies in temperature from one place to another, driving a flow of fluid and heat transfer. The fluid satisfies conservation of mass, conservation of momentum and conservation of energy. In the Boussinesq approximation, variations in fluid properties other than density ρ are ignored, and density only appears when it is multiplied by g, the gravitational acceleration.[1]:127–128 If u is the local velocity of a parcel of fluid, the continuity equation for conservation of mass is[1]:52

where ν (nu) is the kinematic viscosity and F is the sum of any body forces such as gravity.[1]:59 In this equation, density variations are assumed to have a fixed part and another part that has a linear dependence on temperature:

The advantage of the approximation arises because when considering a flow of, say, warm and cold water of density ρ1 and ρ2 one needs only to consider a single density ρ: the difference Δρ = ρ1 − ρ2 is negligible. Dimensional analysis shows that, under these circumstances, the only sensible way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity g′ where

The mathematics of the flow is therefore simpler because the density ratio ρ1/ρ2, a dimensionless number, does not affect the flow; the Boussinesq approximation states that it may be assumed to be exactly one.

One feature of Boussinesq flows is that they look the same when viewed upside-down, provided that the identities of the fluids are reversed. The Boussinesq approximation is inaccurate when the dimensionless density difference Δρ/ρ is of order unity.

For example, consider an open window in a warm room. The warm air inside is less dense than the cold air outside, which flows into the room and down towards the floor. Now imagine the opposite: a cold room exposed to warm outside air. Here the air flowing in moves up toward the ceiling. If the flow is Boussinesq (and the room is otherwise symmetrical), then viewing the cold room upside down is exactly the same as viewing the warm room right-way-round. This is because the only way density enters the problem is via the reduced gravity g′ which undergoes only a sign change when changing from the warm room flow to the cold room flow.

An example of a non-Boussinesq flow is bubbles rising in water. The behaviour of air bubbles rising in water is very different from the behaviour of water falling in air: in the former case rising bubbles tend to form hemispherical shells, while water falling in air splits into raindrops (at small length scales surface tension enters the problem and confuses the issue).